Intersection of Sets Calculator – Find A ∩ B Online

Find the intersection of any two or more sets with our free online intersection calculator. Identifies all common elements shared between sets with clear notation.

Set A (comma or space separated)

Set B (comma or space separated)

Set C (optional)

Examples:

Understanding Set Intersection

The intersection of sets is one of the fundamental operations in set theory. It finds all elements that are common to two or more sets. The intersection of sets A and B is written as A ∩ B and read as "A intersect B" or "A cap B."

Think of intersection as the "AND" operation for sets. An element is in A ∩ B only if it's in A AND in B. This is different from union (A ∪ B), which includes elements in A OR B (or both).

Set Intersection Properties

Definition

A ∩ B = {x | x ∈ A and x ∈ B}

The set of all elements x such that x is in A AND x is in B.

Key Properties

• Commutative: A ∩ B = B ∩ A
• Associative: (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Identity: A ∩ U = A (U = universal set)
• Empty: A ∩ ∅ = ∅
• Idempotent: A ∩ A = A

Worked Examples

Example 1: Basic Intersection

A = {1, 2, 3, 4, 5}

B = {3, 4, 5, 6, 7}

Find A ∩ B:

Common elements: 3, 4, 5

A ∩ B = {3, 4, 5}

Example 2: Disjoint Sets

A = {2, 4, 6, 8, 10} (even numbers)

B = {1, 3, 5, 7, 9} (odd numbers)

Find A ∩ B:

No common elements

A ∩ B = ∅ (empty set)

These sets are disjoint.

Example 3: Three Sets

A = {1, 2, 3, 4, 5}

B = {2, 3, 4, 5, 6}

C = {3, 4, 5, 6, 7}

Find A ∩ B ∩ C:

A ∩ B = {2, 3, 4, 5}

(A ∩ B) ∩ C = {3, 4, 5}

Only 3, 4, 5 appear in ALL three sets.

Example 4: Real-World Application

In a survey of 100 students:

A = students who like Math = {45}

B = students who like Science = {50}

A ∩ B = students who like BOTH = {20}

20 students like both Math and Science.

Quick Fact

Set theory was founded by German mathematician Georg Cantor in the 1870s. His work on infinite sets was revolutionary and controversial. The intersection symbol ∩ was introduced by Giuseppe Peano in 1888, inspired by the logical AND operator.

Frequently Asked Questions

What's the difference between intersection and union?

Intersection (∩) finds elements in BOTH sets – it's the "AND" operation. Union (∪) finds elements in EITHER set – it's the "OR" operation. Intersection makes sets smaller; union makes them larger.

What does it mean if the intersection is empty?

Empty intersection (∅) means the sets have no common elements. Such sets are called "disjoint" or "mutually exclusive." For example, the set of even numbers and the set of odd numbers are disjoint.

Can intersection contain duplicates?

No. Sets by definition contain only unique elements. Even if an element appears multiple times in the input, it appears only once in the intersection. This calculator automatically removes duplicates.

How do I find intersection of more than two sets?

Intersection is associative, so order doesn't matter. Find A ∩ B first, then intersect that result with C, and so on. An element must be in ALL sets to be in the final intersection.

What is the intersection of a set with itself?

A ∩ A = A. Every element in A is obviously in A, so the intersection contains all elements of A. This is called the idempotent property of intersection.

Where is set intersection used in real life?

Database queries (SQL JOIN), search engines (finding pages matching multiple criteria), filtering products by multiple features, Venn diagrams in statistics, and probability (P(A and B) uses intersection).

Other Free Tools

Search tools